Atoms are about as small as anything gets, at least in any terms the average person can comprehend. Any smaller, and you’re looking at a particle that no longer retains any of its chemical properties: You can’t have, for instance, half an oxygen atom, without it ceasing to be oxygen any more. The size varies a little, from the smallest (Helium, at around 32 picometres), to the largest (one of which is caesium, at 225pm). A picometre, in case you’re wondering, is 1/1,000,000,000,000th of a metre, or one millionth-millionth.
Atoms are composed of three main parts. Positively charged protons and neutrally charged neutrons make up the nucleus of the atom, around which in various orbits are negatively charged electrons. In most atoms, there are a roughly equal number of protons and neutrons; in standard carbon atoms, for instance, there are six protons and six neutrons. Carbon 14, familiar as the isotope used in the Carbon Dating method, is radioactive, and has an uneven amount; 6 protons, and 8 neutrons. It is this imbalance which gives the particle its radioactivity. Typically, an atom has the same number of protons and electrons also, giving it a neutral charge, but some have a deficit or surplus of electrons, leading to what we know as ionised particles.
The number of protons/neutrons is what makes an atom what it is. Conceivably, if you could add additional protons and neutrons to an atom (or take them away), you would change its atomic number altogether, you would change the element entirely, giving us the ancient alchemical dream of transmutation in a nutshell. This has been done, but is so prohibitively expensive, it’s really not worth it. Lead /can/ be changed into gold, but requires far more energy than the gold would actually be worth…
Most people envision the atom as a dense core, surrounded by rapidly swirling electrons, like a tiny solar system of planets orbiting a sun. This is true, to a certain extent, though what with Quantum Theory and all, it’s impossible to actually say where a particular atom is, and if we could, we would disturb its position anyhow. Thus for scientific purposes, the electrons are considered more of a ‘cloud’, a distribution of probable locations where the electrons /might/ be. Given that they can apparently jump from one location to another without crossing the intervening space, this is probably for the best. The ‘tiny solar system’ model is about as comprehensible as the structure is likely to get for most of us.